Evolution of frequency clusters in the naturally occurring dust acoustic wave

The spatiotemporal evolution of the naturally occurring dust acoustic wave mode is experimentally investigated in a weakly coupled dc glow discharge dusty plasma system over a range of neutral gas pressures through the application of a time-resolved Hilbert Transform. Frequency clusters are observed over a range of neutral gas pressures, though their spatial distribution varies with neutral gas pressure. It is also observed that the wave frequency is observed to drop by $\sim 10$ Hz across these frequency clusters independent of the experimental parameters.

Figure 1

Sketch of the experimental apparatus. The dust clouds examined here form between the anode and the electrically floating dust tray.

Figure 2

(a) An inverted image of a dust cloud at $p$ = 91 mTorr. (b) A space-time diagram constructed from intensity profiles extracted at $x$ = 9 mm in the image seen in (a). The light bands denote the propagation of individual wavefronts through the dust cloud. The (yellow) dashed-dot lines indicate the top and bottom of the dust cloud.

Figure 3

(a) Plot of the wave structure at a single pixel location as a function of time (solid, red curve). The dashed (blue) curve denotes the envelope function. (b) Plot of the unwrapped phase as a function of time at the same pixel location. (c) Plot of the mean frequency (solid, black curve) as a function of time superimposed on a periodogram constructed by applying a short-time Fourier Transform to the measured wave structure. The sudden variations that are observed in the mean frequency occur when bifurcations are observed on the wave structure and are correlated with discontinuities in the unwrapped phase and sudden decreases in the wave amplitude, suggesting the existence of a topological defect.

Figure 4

Comparison of the time-resolved Hilbert transform and the Fourier transform at the three spatial locations indicated in the image of the dust cloud in the left column. The middle column compares the time-resolved Hilbert transform (solid, black curve) to the frequency found by performing a short-time Fourier transform on the measured wave structure over the entire measurement sequence, while the right column depicts the distribution of the the mean frequencies found from the application of the time-resolved Hilbert transform (dashed, blue curve) and the Fourier spectra found by applying the Fourier transform to the wave structure over the entire measurement sequence (solid, red curve).

Figure 5

Plot showing (a) an inverted image of the dust cloud, (b) the instantaneous phase, and (c) the spatial distribution of frequency at a single time step with $p$ = 102 mTorr. The phase and frequency plots have been filtered to only show points where wave activity was observed. Contours showing where the standard deviation in a 5 $\times$ 5 pixel region exceed 1 Hz are superimposed on the spatial distribution of frequency and serve as an indication of a boundary between frequency clusters. Movies showing the time evolution of this system at different pressures are available in the Supplemental Material [41].

Figure 6

Image sequences showing the temporal evolution of the (top row) wave structure and (bottom row) spatial distribution frequency at $p$ = 91 mTorr (left column) and 102 mTorr (right column). Contours depicting where the standard deviation in a 5 $\times$ 5 pixel region exceed 1 Hz are superimposed on the spatial distribution of frequency, which has been filtered to only show regions where there is significant wave activity. Arrows have also been added to highlight the propagation of particular wavefronts. The time between images is 40 ms.

Figure 7

Space-time plots constructed by extracting a slice of the (left column) acquired image and (right column) spatial distribution of frequency from the center of the dust cloud at $p$ = 91 mTorr (top row), 102 mTorr (middle row), and 118 mTorr (bottom row). Contours depicting where the standard deviation in a 5 $\times$ 5 pixel region exceed 1 Hz are superimposed on the space-time plots in the right column, which has been filtered to only show regions where there is significant wave activity.

Figure 8

Plot of the frequency after a frequency cluster, $f_{\mathrm{lower}}$, as a function of the frequency before a frequency cluster, $f_{\mathrm{upper}}$, at the different neutral gas pressures examined. A linear fit (black curve) applied to all of the measured frequency pairs indicates that the frequency drops by $\sim$10 Hz across these frequency clusters independent of the experimental parameters.